A Bayesian Perspective on Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function

Pollard used contour integration to show that the Mittag-Leffler function is the Laplace transform of a positive function, thereby proving that it is completely monotone. He also cited personal communication by Feller of a discovery of the result by ''methods of probability theory''. Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the result. We prove the result by a Bayesian approach. We proceed to prove the complete monotonicity of the multi-parameter Mittag-Leffler function, thereby generalising the Pollard result by methods of Bayesian probability theory.